Oversize Tubular Metallic Waveguides*

3.2

Oversize Tubular Metallic Waveguides* John P. Quine

I. Introduction . . . . . . . . . . II. Tubular Metallic Waveguides as High-Power Transmission Media. A. Waveguide Attenuation Caused by Finite Wall Conductivity . B. Mode Conversion Coefficients for Typical Discontinuities c. Waveguide Temperature Rise with High Average Power and Free Convection Cooling . . . . . . . . D Dependence of Peak Power Carrying Capacity on Waveguide Configuration and Mode Type . . . . . . E. The Effect of Spurious Mode Resonances . . . . III. Design of Oversize Waveguide Components for High-Power Systems A. Design of Tapers . . . . . . . . B. Design of Bends . . . . . . . . C. Design of Multihole Directional Couplers . . . . D . Design of Quasi-Optical Couplers E. Additional Components IV. Conclusions . . . . . Symbols . . . . . References . . . . .

178 180 180 183 185 186 187 189 190 198 201 207 208 209 210 211

I. Introduction

This section is concerned with the efficient transmission of high microwave power by means of a waveguide consisting of a highly conducting hollow cylinder or tube having arbitrary but uniform cross-sectional shape and filled with a low-loss dielectric (usually a gas). In particular, consideration is given to the use of oversize waveguides having cross-sectional dimensions that are large compared to the free space wavelength, λ. Power can be transmitted along the inside of a hollow conducting tube in a particular mode only if the operating wavelength is less than the characteristic modal cutoff wavelength determined by the internal cross-sectional shape and dimensions of the tube [1,2]. The two most common cross-sectional shapes employed in practice are rectangular and circular. In the case of a rectangular waveguide, the width, a, must be at least 0.5Λ for propagation to be possible * Part of the work covered in this review was performed under Air Force Contracts AF30(602)-2900 and AF30(602)-3682 with the Rome Air Development Center, Griffiss Air Force Base, New York. 178

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in the TE° 0 mode,| the lowest-order mode for a rectangular waveguide; for this mode, the cutoff wavelength is independent of the waveguide height, b. In the case of a circular waveguide, the diameter, Z>, must be at least 0.586Λ, for propagation to be possible in the TEp 1 mode, the lowest-order mode for a circular waveguide. Higher-order modes can also propagate, if the waveguide dimensions are large enough relative to λ. For example, the T E ^ and T M ° modes that are degenerate (have the same cutoff wavelengths) can propagate in a square waveguide if a is equal to or greater than 0.7072, and the TM^i mode can propagate in a circular waveguide if D is equal to or greater than 0.7652. The circular electric TE^i mode and its degenerate partner, the T M ^ mode, can propagate in circular waveguide provided D exceeds 1.22A; this degeneracy creates a problem in designing a bend for the ΤΕ£Ί mode [3]. It should be noted also that many of the modes of square and circular waveguides, including the TE° 0 and TEft modes, have cross-polarized degenerate mode partners [1]. Minor cross-sectional irregularities can, therefore, cause linear polarization to transform to elliptical when transmitting these modes over long distances [4]. The design of a waveguide system is simplified if the waveguide crosssectional dimensions are made small enough to allow only the lowest-order mode to propagate (without cross-polarized degenerate modes). In this case, energy loss can occur only as a result of dissipation and reflection. On the other hand, if more than a single mode can propagate, energy loss can occur also as a result of the conversion of energy from one propagating mode to another. For these reasons, the commonly employed standard rectangular waveguides have a ratio, a/b, approximately 2 and width, a, in the range between approximately 0.65 and 0.9Λ.. Restricting the waveguide size to allow only the lowest-order mode to propagate, however, limits the peak and average power-handling capability of the waveguide. Furthermore, the attenuation of a long run of standard rectangular waveguide due to dissipation is often much higher than can be tolerated from an economic standpoint. For example, the attenuation of standard RG 52/U rectangular waveguide operating at 10 GHz is of the order of 4 dB/100 ft, and the peak power is limited by air breakdown to approximately 0.3 MW without pressurization, whereas the average power is limited to approximately 5 kW for 150°C temperature rise with natural convection air cooling without fins [1,5,6]. The power-handling capability of microwave systems can be increased by orders of magnitude through the use of oversize waveguides and components. One proposed application is the transmission of super-high microwave power (gigawatts) for long distances [7] ; in this case, the use of the TEjp, mode in t In this section the superscripts(D) and (O) designate rectangular and circular waveguide modes.

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JOHN P. QUINE

circular waveguides having a diameter equal to approximately \0λ can result in several orders of magnitude increase in power-handling capability relative to standard waveguide. For relatively short waveguide runs, the use of the TE° 0 mode in rectangular waveguides having both cross-sectional dimensions equal to approximately 2λ can result in more than a single order of magnitude increase. However, in order to realize these increased power-handling capabilities, the conversion of energy from the desired mode to undesired or spurious modes must be minimized. In addition to transmission loss for the desired mode, mode conversion can also lead to a reduction in power-handling capability as a result of resonance caused by trapping of the energy converted to a spurious mode [8,9]. In the following sections, the effect of finite wall conductivity, mode conversion, and spurious mode resonances on waveguide attenuation and peak and average power-handling capabilities will be reviewed, and the design of components having low mode conversion loss will be discussed. II. Tubular Metallic Waveguides as High-Power Transmission Media A. WAVEGUIDE ATTENUATION CAUSED BY FINITE WALL CONDUCTIVITY

Figure 1 shows the normalized theoretical attenuation [1] resulting from wall losses plotted as a function of the normalized waveguide dimensions. The quantity, A, plotted as the ordinate, is the ratio between the attenuation od, in decibels, occurring in one free space wavelength, λ, and the normalized surface resistance, ϋ5/η9 of the waveguide walls; a is the attenuation in decibels per unit length, Rs is the surface resistance of the waveguide walls, and η is the free space impedance. In order to obtain the attenuation in decibels in one free space wavelength for a particular combination of frequency and surface resistance, A obtained from Fig. 1 must be multiplied by Rsfa given by /?s/f/ = 6 . 9 2 x l O - 1 0 ( / K ) 1 / 2 ,

(1)

where/is the frequency in cycles per second and σΓ is the conductivity relative to the dc conductivity of copper (5.8 x 107 mho/m). The solid curves in Fig. 1 give the value of A for the TEfJ, mode in rectangular waveguides. The curve for b/a = oo represents the attenuation for an infinite parallel-plate waveguide with the electric field parallel to the plates ; in this case, the losses are due entirely to transverse side wall currents. The curve for b/a = oo also represents the attenuation for the type of transmission line known as "//-guide" [10] for vanishing dielectric thickness. The difference in decibels between the curve for a particular finite value of b/a and the curve for b/a = oo corresponds to the attenuation in decibels due to top and bottom wall currents.

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OVERSIZE TUBULAR METALLIC WAVEGUIDES

0.6 0.8 I

2

4 α/λ Or D/λ

6

8

10

20

FIG. 1. Normalized waveguide attenuation as a function of waveguide dimensions.

The two dashed curves give the values of A for the TEft and TE£*i modes in circular waveguide. Note that, when D/λ exceeds approximately 1.3, a circular waveguide of diameter, D, supporting the TEfr mode has a slightly lower attenuation than a square waveguide of side dimension, Z), supporting the TE^ 0 mode. The attenuation for the TE£i mode is greater than that for the TEfi mode or for the TE° 0 mode in square waveguide of side dimension, D, for values of D/λ less than approximately 1.8. For very large normalized waveguide dimensions, the normalized attenuation, A, is approximately proportional to λ/α for the TE? 0 mode for finite b/a, to λ/D for the TEft mode, to (λ/D)3 for the TE& mode, and to (λ/α)3 for the TE° 0 mode for b/a = oo.

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JOHN P. QUINE

The reference point for standard waveguide shown in Fig. 1 has a value of A which is equal to 42. The curves show that a single order of magnitude reduction in loss, which requires A equal to 4.2, can be obtained by employing square or circular waveguide having cross-sectional dimensions in the range between 2 and 2.5 wavelengths. The curves also show that in this range of waveguide dimensions the TE§\ mode does not have significantly lower loss than the TE° 0 or ΊΕγι mode. In order to obtain two orders of magnitude reduction in attenuation, A must equal approximately 0.42. For this, the TEjpi mode with D/λ equal to approximately 4 can be used. The TEft mode or the TEfo mode with b/a = 1.0 would require cross-sectional dimensions equal to approximately 20 wavelengths. If b/a is equal to five or more (tall waveguide), use of the TE° 0 mode with α/λ ranging between four and five would also result in two orders of magnitude reduction in loss. It has been shown recently [9,11,12] that the usual power-loss method [1] of calculating attenuation due to wall losses can lead to substantial errors in the case of degenerate modes even for very small attenuation values. (In the power-loss method, the attenuation is calculated by assuming that the fields and wall currents in the lossy waveguide are approximately the same as in the loss-free waveguide.) In the case of a rectangular waveguide, it has been shown that the TE°„ and TM°„ modes that are degenerate characteristic modes of a loss-free waveguide are coupled by a finite value of Rs and are therefore not the stable modes (field patterns preserved at all cross sections along the axis) of a lossy waveguide. Instead, the two stable modes of order m, n of the lossy waveguide are linear combinations of the degenerate TE°„ and TM°„ modes of the loss-free waveguide. Furthermore, the attenuation constants of the stable modes of order m, n differ substantially from the attenuation constants of either the TE°„ or TM°„ modes, as calculated by the power-loss method. In fact, the two stable modes of order m, n represent the two linear combinations of the TE°M and TM°„ modes which give maximum and minimum attenuation as calculated by the power-loss method. Far above cutoff, the two stable modes tend to the longitudinal-section E modes polarized at right angles to each other in the transverse plane. The power-loss method gives the correct result in the case of the circular waveguide TE^i and TEft modes, because these modes are stable modes with or without loss. Although degenerate, the TEp t and TMft modes are not coupled by the surface resistance. Summarizing, it is seen that the TE? 0 and TEft modes should be considered for use in moderately oversize waveguide systems (transverse waveguide dimensions of the order of two wavelengths). As will be clear from later discussions, these modes have significant advantages over the TE£\ mode in the design of certain important components. For transverse dimensions considerably greater than two wavelengths, the TE§\ mode has the advantage of much

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lower loss. Long-distance transmission of high power using the TE(?, mode has been discussed by Okress [7], and is considered by Dunn and Loewenstern later in this volume. B. MODE CONVERSION COEFFICIENTS FOR TYPICAL DISCONTINUITIES

When oversize waveguides are employed, losses can occur as a result of the conversion of energy from the desired mode to other propagating modes. Any departure from the perfect cylindrical geometry causes coupling among the cylindrical waveguide modes, and results in mode conversion losses. Mechanical imperfections or discontinuities can be classed as either discrete or continuous. Examples of discrete discontinuities are abrupt axial misalignments or offsets, abrupt axial tilts, and abrupt changes in the waveguide crosssectional dimensions. Gradual offsets, bends, and tapers are examples of continuous discontinuities. Rowe and Warters [13] have discussed the transformations from the discrete to the continuous case, and the relationships between the various coupling coefficients. The normalized amplitude coupling coefficient, C, for the abrupt offsets, tilts, and diameter changes shown in Fig. 2 are listed in Tables I—III. With

1 T

t

s

J - r -_i_

'

/■

1

/

*

T

~r

V

\/

-r—- Λ\ lb) Tilt

(a) Offset S

_i

-

^

r L

~T S (C)

FIG.

Diameter Change

2. ,Abrupt discontinuities.

unit incident power in the desired mode, the coupled power is equal to C 2 , provided the spurious mode is terminated in a matched load (see discussion of spurious mode resonance later in this section). These formulas, which were obtained from several sources [13,14-16], apply for modes that are not too close to cutoff. The factors knm occurring in the formulas for circular waveguide are the roots of Jn'(knm) = 0. Thus, k0l = 3.832, kn = 1.841, kl2 = 5.330, k02 = 7.016.

184

JOHN P. QUINE TABLE I MODE CONVERSION IN ABRUPT OFFSETS

Incident mode

Coupling coefficient C

TE8 (T=D)

2v2(-)

TEpx (H plane) (T=D)

2V

TE?o (E Plane) (T=b) Τ Ε ^ ( # plane) (T=a)

k

Z

\D)(k2m-k201)(k2m

-^

Spurious modes

koikU

*(f)ë«T - * o i ) ( « i - l ) " V 2AsUsinC«77S/6)"| (mrS/b)

t

TEä

l) 1

WN

4m

ΤΕθ

2

Composite T E H / T M S

f)

for all n

TES) modes for even m only

TABLE II MODE CONVERSION IN ABRUPT TILTS '

Coupling coefficient C

Incident mode TES (Γ=Ζ>)

TEfi (T=D)

(*8ι-*ϊ«Χ*ϊ,.-ΐ)"2 I * / W

TE?o (E plane)

TEPO (H plane) (Γ=β) a

2^277^0,^5,

4

-£(;)(ί)

16m M M (m2 - l)2 W \π)

Θ is tilt angle expressed in radians.

TEpm

TMfi

*0 V2 \ A/ \ w ;

ΤΕβ (// plane) (Γ=Ζ))

Spurious modes

/ ^ V ^

TE8

Composite TE?JTM?„ for odd n only TES) for even m only

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OVERSIZE TUBULAR METALLIC WAVEGUIDES TABLE III

MODE CONVERSION IN ABRUPT DIAMETER CHANGES

Incident mode

Coupling coefficient C

TEfi

AS

ko\kQm

D,

2

(T=D)

TEfi

AS

TE?o(£- plane)

iVis

TES (H plane)

b2

TEfi

L· l·2

TMfi

(T=D)

(T=b)

Spurious modes

sm{nnSlb2) _ (nrrS/b2)

2m[(a2/a1)-\] m2 - (a ,/a,)2

Composite T E E / T M B for even n only TES, for odd m only

In the case of uniform continuous coupling (or with identical close-spaced discrete couplings) between two modes, Miller [17] has shown that power transfers cyclically between the two modes along the length of the coupling region. Coupling coefficients for continuous coupling can also be found in the literature [13,18-21]. The case of random imperfections has also been discussed in detail [13]. The theoretical coupling coefficients are useful for component design and in establishing dimensional tolerances. For example, use of the formula in Table IT for an abrupt £-plane tilt in rectangular waveguide shows that, for 6/^ = 2.0, 1° tilt results in approximately —24 dB coupling between the TE^o mode and the composite T E ° / T M ° modes. In this case, the composite modes (known also as LSE modes [9,11] are combinations of the TE°„ and TM°„ modes, and have transverse electric field parallel to the electric field of the incident TE° 0 mode. C. WAVEGUIDE TEMPERATURE RISE WITH HIGH AVERAGE POWER AND FREE CONVECTION COOLING

The amount of power that can be transferred per unit length in a waveguide run cooled mostly by free air convection (radiation negligible) is proportional to the heat-transfer coefficient for free air convection of the outside surface. In this case, the temperature rise, AT, is given by [6] AT{°C) = WISH, where W = watts dissipated per foot of waveguide S = surface area in square inches per foot of waveguide H = heat-transfer coefficient for free air convection (W/in.2-°C)

(2)

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JOHN P. QUINE

As an example, consider a length of 2.8 x 2.5 in. copper rectangular waveguide operating at 9 GHz and carrying 100 kW of average power. In this case, S = 2 x (2.8 + 2.5) x 12 in. = 127 in. 2 . The theoretical attenuation for this waveguide can be obtained from Fig. 1 and Eq. (1), and is approximate./ 0.0034 dB/ft. This results in a dissipation of 68 W/ft for 100 kW carried by the waveguide. The temperature rise, AT, is therefore AT =

68 = 81°C, 127 x 0.0066

where a calculated value of 0.0066 W/in.2-°C [6] has been employed for the heat-transfer coefficient, H. Measured temperature rises under the foregoing conditions have ranged between 110° and 140°C, indicating that the actual losses may have been somewhat higher than theoretical. The temperature rise for a given heat dissipation can be reduced substantially through the use of cooling fins. With fins, the heat transfer for a given length of waveguide can be easily increased by a factor of 5. Assuming a factor of 5, the temperature rise in the previous example would be reduced to approximately 30°C. Horizontal runs of waveguide may be cooled with relative ease, since the vertical cooling fins can work most effectively. Angled fins can be employed on vertical waveguide runs. D. DEPENDENCE OF PEAK POWER CARRYING CAPACITY ON WAVEGUIDE CONFIGURATION AND MODE TYPE

The following equations [5] give the peak breakdown power, P m a x , as a function of the peak breakdown field intensity, Emax : For the TE? 0 mode, ^max(kW) = 0.662 x 10" 6 abßßg)E2max.

(3a)

For the TEPX mode, U k W ) = 0.498 x 10" 6 D W , ) E £ a x .

(3b)

For the TEjpi mode, Pmax(kW) = 0.501 x 10" 6 ϋ\λΙλ9)ΕΐΆΧ.

(3c)

λ and Xg are the free space and waveguide wavelengths, respectively. Equations (3a)-(3c) can be written in the following normalized form: i>max(kW)M(cm)2 = F[£ max (V/cm)] 2 ,

(4)

where A is the cross-sectional area and F is a factor proportional to λ/λ9. For air at atmospheric pressure and 20°C, Emax is approximately 2.9 x 104 V/cm.

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The maximum power per unit area for this case is plotted in Fig. 3. The advantage of the TE° 0 and TEft modes over the TEfi mode for waveguide dimensions less than approximately 2λ should be noted. As an example, assume a rectangular waveguide having a = b = 2λ = 7.5 cm (8 GHz); in this case, Pmax = 540 x (7.5)2 = 30 350 kW = 30.35 MW. 600 556 λ / λ ς

TE?o 500

^

400

532 λ / λ ς

TE£^

535 λ / λ ς

TrO

/

E

O

5· 30 ° 0_

200 EMAX

= 2.9 xlO 4 Volt s/cm

100

0 10

1.5

2.0 α/λ Or D A

2.0

3.0

FIG. 3. Peak power-handling capability.

E. THE EFFECT OF SPURIOUS MODE RESONANCES

Resonance may occur when an undesired or spurious propagating mode is trapped within a region of the oversize waveguide system, e.g., in the region between two tapered transitions. When resonance occurs, large amounts of power may be converted from the desired mode to the spurious mode; in this case the spurious mode fields may build up to high values, and overheating and dielectric breakdown may occur. Mode-selective absorbers can be employed in order to prevent high field buildup; these can be designed to provide high absorption for the spurious mode and negligible absorption for the desired mode [22].

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JOHN P. QUINE

The fraction of the power which is converted (and dissipated) when both the desired and spurious modes are terminated in their characteristic impedances is equal to C 2 , where C is the normalized amplitude coupling coefficient between the desired and spurious modes ; this condition is approached when a mode-selective absorber having high absorption is employed. With finite absorption, however, the fraction of the power which is dissipated can be considerably higher or lower than C 2 , depending on whether or not the trapped spurious mode is resonant. Calculations based on an approximate three-port equivalent circuit have been made for the case of a gradually tapered transition from a single-mode waveguide to an oversize waveguide [8,9]. In this case, it is assumed that the single-mode waveguide input port (port 1) is matched when the oversize waveguide is matched for the spurious mode (port 2) and for the desired mode (port 3). The ratio between the power, P 2 , dissipated in the trapped spurious mode and the power, Pi9 incident in the desired mode has a maximum value at resonance given by /ΡΛ Wres

=

4C 2 cothQ4/8.686) [2 - C 2 + C 2 coth04/8.686)]2 *

C }

In deriving (5), it was assumed that a short-circuited line having one-way loss equal to A (expressed in decibels) was connected to port 2, with a matched load on port 3. The maximum voltage standing wave ratio for the desired mode occurs at resonance and is given by (VSWR)res = 1 - C 2 + C 2 coth(A/8.686).

(6)

By an extension of Greimsmann's analysis [8], it can be shown that the maximum value of the ratio between the normalized amplitude, A2, of the electric (or magnetic) field of the trapped spurious mode and the normalized amplitude, A1, of the electric (or magnetic) field of the incident desired mode occurs at resonance and is given by

a,=[—(^)r'"C'

where the ratio, Ρ 2 /Λ» *s obtained from (5). The ratio (A2/Ai)res represents the factor by which the normalized amplitude of the electric (or magnetic) field is built up as a result of the resonance {\A2IAX\2 is effective power buildup). Equation (5) shows that (P2/P1)res n a s a maximum possible value equal to 1/(2 - C2) when the condition 2 - C2 = C 2 coth(^/8.686) is satisfied; this defines a worst coupling condition. For small values of A and C, the condition for worst coupling reduces to C2 = AJ4.343. Note that, for the worst coupling

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condition, CP2/^i)res ls equal to 1.0 when C is equal to 1.0, and is equal to 0.5 when C is small compared to 1.0. Equation (6) shows that under the worst coupling condition the value of (VSWR)res is equal 3 - 2C 2 ; this is equal to 1.0 when C is equal to 1.0, and is equal to approximately 3.0 when C is small compared to 1.0. Equation (6) also shows that, for C 2 much less than Λ/4.343, the value of (VSWR) res is nearly equal to unity; that is, the spurious mode resonance causes only a small impedance mismatch under these conditions. It was shown previously that the value of(P2/Pi)res ranges between 0.5 and 1.0 under the conditions of worst coupling. Therefore, (7) shows that the highest possible value of A2/Al is within 3 dB of an upper bound, given by [1 — exp( — 4Λ/8.686)]" 1/2 ; for small values of A, this is equal to approximately (8.686/4Λ)1/2. The buildup of the electric and magnetic fields occurring when a trapped spurious mode resonates can cause the actual average and peak power-handling capabilities of the waveguide system to be considerably lower than those predicted by Eqs. (2)-(4). The reduction in power-handling capability can be avoided, however, through careful design to minimize the mode conversion coefficient, C, and through the use of mode-selective absorbers with sufficiently large values of spurious mode adsorption, A, to dampen the resonance buildup [22]. As an example, consider the typical case, C = 0.1 or —20 dB; the worst coupling condition requires A approximately 0.043 43 dB. Under these conditions, (iV^i)res *s approximately 0.5, and (A2IAl)res is approximately 14.7. On the other hand, if a value of A = 3.0 dB is employed with C = 0.1 (5) gives ( / y / U e . = 0.0295 and (7) gives (AJA,)^ = 0.2. III. Design of Oversize Waveguide Components for High-Power Systems

In the following sections, the design of oversize waveguide components is discussed. One class of components which is considered has moderately oversize cross-sectional dimensions ranging between 1.5 and 2.5 free space wavelengths, λ. In this class of components, the spurious propagating modes are relatively few and have widely varying characteristics. Advantage can often be taken of the differences in the symmetries of the field distributions of the individual modes in order to avoid the excitation of certain of the undesired modes when the desired mode is incident on a waveguide discontinuity. Advantage can also be taken of the differences in the propagation constants of the individual waveguide modes. For cross-sectional dimensions in the foregoing range, these differences can be substantial, and the spurious modes can be suppressed by a "phasing-out" process [17], provided the coupling to the spurious modes takes place over an axial distance comparable to one or more beat wavelengths.

190

JOHN P. QUINE

A second class of components which is considered has cross-sectional dimensions that are 1(M or greater. For these components, the number of propagating modes becomes extremely large, with a great many modes having nearly the same propagation constant and field symmetries as the desired mode. Therefore, the preceding principles alone cannot be employed to obtain low mode conversion loss. For this class of components, however, the principles of quasi-optics [23,24| can be applied. The application of the foregoing principles will be apparent in the following discussion of oversize waveguide components such as tapers, bends, and directional couplers. These components can have power-handling capabilities that are orders of magnitude greater than those of standard-size waveguide components. The principal design problem is the reduction of the spurious mode amplitudes to acceptable levels ( — 25 to —30 dB relative to the incident desired mode). A. DESIGN OF TAPERS

Tapers are required to transform from the standard-size waveguide outputs usually employed for microwave power sources to the oversize waveguide transmission line. Tapers may also be required in certain oversize waveguide components to transform dimensions to values required for optimum component design. One can design either straight tapers (Fig. 4a) with constant Abrupt Discontinuities

Output
(o) Straight Taper

Input d,

Output d 2

(b) Variable Taper F I G . 4. Waveguide tapers.

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taper angle and abrupt discontinuities at each end, or variable tapers (Fig. 4b) with taper angle varied continuously along the length of the taper. 1. Straight Tapers Solymar has analyzed straight tapers for circular waveguides carrying the TE^i mode [25,26-28], and for rectangular waveguides carrying the TE° 0 mode [29]. In the case of circular waveguides of diameter, Z), carrying the TE^i mode, the tapers generate the TE$, spurious modes. If the input waveguide in Fig. 4a is below cutoff for modes having n > 1, mode conversion loss occurs only at the junction with the output waveguide. The normalized amplitude coupling coefficient, C, between the TEf*i and TE£2 modes at the output junction is given by [25] C = 0.142

2V

2

-,

(8)

AL·

where Z)x and D2 are equal to dx and d2, respectively (Fig. 4), and C 2 represents the power coupled to the spurious mode for unit power incident in the desired mode. In the case of rectangular waveguides having width, a, and height, b, and carrying the TE^ 0 mode, the odd-order TE° 0 spurious modes are generated in a symmetrical //-plane taper, whereas the even-order composite TE°„/TM°n or (LSE) ln spurious modes are generated in a symmetrical £-plane taper. The amplitude coupling coefficient, C, between the TE° 0 and the TE° 0 modes in a symmetrical //-plane taper (di = al; d2 = a2) is given by [29] C = 0.375

fl2(fl2 ul)

nAL

"

(9)

The amplitude coupling coefficient between the TEJ^ mode and the composite TE 12 /TM 12 mode in a symmetrical £-plane taper (d{ = bl; d2 = b2) is given by [29] b2(b2 - bt) r (10) nXL ' Comparison of (9) and (10) shows that the coupling to the composite TE° 2 /TM° 2 mode in an £-plane taper is nearly 10 dB stronger than the coupling to the TE° 0 mode in an //-plane taper of comparable dimensions. Solymar's results also show that the coupling coefficients decrease rapidly with the order of the spurious mode. In particular, the next higher-order mode coupling coefficients are approximately 10 dB lower than the values for the lowest-order spurious modes given by (8)-(10). Note that the amplitude coupling coefficients for straight tapers are inversely proportional to the taper length, L ; in this case, doubling L reduces mode conversion by only 6 dB,

192

JOHN P. QUINE

and inconveniently long lengths may be required to obtain a very low specified mode conversion with a straight taper. 2. Variable Tapers Very low mode conversion can be obtained with reduced taper lengths by employing a variable taper (Fig. 4b). In this case, the taper angle is varied to produce a specified mode conversion distribution [18,30] along the length of the taper. One choice for this distribution is 8ΐηη(πρ/ρ1), where the exponent, n, can have an arbitrary value, and the factor, p\px, is a normalized phase parameter that ranges between zero and one along the length of the taper. The design equations for variable tapers have been derived elsewhere [18,30,31]; only the results of these derivations will be given here. The design equations give the axial distance, z, along the taper at which the variable transverse dimension has a value, d. Both z and d depend on the phase factor p/pl. The equations for z can be written in the following form: (11)

z = K1F(z9x)-K2n(plp1).

In (11), the factor F(a, x) is the same for all waveguide modes considered here, and depends only on the exponent, n. The factors Kx and K2 are the same for n = 1, 2, and 3, but depend on the waveguide mode. The factor F(a, x) is given by the following equations : F(a, x) = 1 +

+

a1 7

8

+

+

+

a* /15\ a0 64 + \48/ 720 128

+

\450/

\96/ 720J sin x +

5 Ί

a sin x Ï5Ô

a + sin 2x \24/ 720

+ 96

— -—cos x sin x H cos x sin x 4320 600 F(a, x) =

cos x sin x for

n = 1,

(12)

1 dv fexp(ax) — 1

Yd, 1-2

-I[a exp(ax)sin x — exp(ax)cos x + 1]

a 2 exp(ax) /a cos 2x + 2 sin 2x\ a2+ 4 (-


+ 4(ÎM) + i C e x p ( a x ) - i : i

for

2,

(13)

3.2

F(a, x) = x

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OVERSIZE TUBULAR METALLIC WAVEGUIDES

alsin x + - sin3 xl 123 123 31 - ^ x + - ^ sin 2x — — cos 3 x sin x + - cos5 x sin x 48 96 24 6

+ 48

- 1 0 .962lsin x 7 3

6

·

sin3 xj + 4.010 cos x sin x 1

8

-

"

cos x sin x + - cos x sin x 63 9

3 23.106Î-«cos x sin x + — sin 2x + 16

+ 384

■

)

-

11.580 cos5 x sin x + 5.503 cos7 x sin x 133

. 1 n . 9 cos x sin x H cos x sin x 120 12 4 48.61 · 3 \ cos 4 x sin x + 4 sin x + 3840 sin·3 x) 3 / 200.48 226.79 cos x sin x — cos x sin x +

~T

+

76.46 11

cos 10 x sin x — 1.128 cos

x sinx + — cos 14 x sin x for

n = 3.

(14)

In (12) and (14), a = In d2\dx and x = np/pt ; in (13), a = (l/π) In d2/di and x = 2πρ/ρι. For all n, dt and d2 are the values of d for the input and output waveguides, respectively. Equations ( 12)—( 14) are approximations obtained by the series expansion of an integrand using terms up to the sixth, third, and fifth powers, respectively.* For circular waveguides carrying the TE^i mode, for n = 1, 2, and 3, Ki =

4/?o Q| a2 Pi

n{k\ - k\) '

K, =

(*i + k\)Px

n{k\ - kDßo'

(15)

* Equations (12) and (14) are more complete than the corresponding equations given by Tang [30] which contain several typographical errors; Eq. (13) agrees with Eq. (24) by Unger [18] when p is set equal to px.

194

JOHN P. QUINE

where al and a2 are the radii of the input and output waveguides, respectively. The km are the mth roots of the Bessel function, J1, of the first kind, and ß0 is 2π/λ. px is the value of p at the output of the taper, and depends on the length selected for the taper. The following equations apply for rectangular waveguides of width, a, and height, b, for n = 1, 2, and 3 : (a)

//-plane taper (a varied, b fixed): K,=

(b)

5 Pi 4 *

K,=-

2π 3

(16)

£-plane taper (a fixed, b varied): [4/?ο-(2π2/0οα2)]Μ2Ρι Απ'

X, =·

K,=

πβο'

(17)

(c) Composite E-H plane taper (both a and b varied, alb fixed). In this case, one usually designs the £-plane taper: „

2b, b2 π λ0

2+

fë)'

(18)

The variable cross-sectional dimension, d, of the waveguide is given by the following equations, which apply for all modes :

dl d ~dx

d

exp

é

exp

^= expu

1 P Pi

1

cos

©

ln-p «1

for

1 . 0 P sin 2π — l n ^ l Pi 2π

(19)

n = l, for

H^B»^)

(20)

n = 2, for

n = 3. (21)

«1

The factor, p,, occurring in the equations for A\ and K2 can be determined from the following equations, which apply for all modes:

al a ai c C

= (cos p^

-

&

for

n = 1,

(22)

for

n = 2,

(23)

(24)

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In (22)-(24), A2 is the normalized spurious mode amplitude at the output of the taper,* and C is a coupling factor that depends on the waveguide mode. For circular waveguides carrying the ΤΕ£Ί mode, C = ^ 4 In ^ = 1 . 5 5 In ^ . k2 — ki a{ ax For rectangular waveguides carrying the TE° 0 mode,

(25)

r b2 C = J2 In -=

for £-plane tapers,

(26)

C = 0.75 In —

for //-plane tapers.

(27)

bi

The minimum value of ργ required for a specified ratio of d2/dl and a maximum allowable value of A2 can be determined from (22)-(27). The minimum taper length, L, can then be determined from (11) by setting p = pv A study of (22)-(24) shows that, for a given value of \A2/C'\9 an optimum value of n exists which results in minimum pl, and therefore minimum taper length, L. Usually, the higher the allowable mode conversion, the lower the optimum value of n. In some narrow-band applications in which relatively large mode conversion can be tolerated, a multisection straight taper [26,27] may have a shorter length than a variable taper having the same mode conversion. The cross-sectional dimensions for composite E-H plane rectangular waveguide variable tapers are given in Tables IV and V. These data were obtained from computer calculations using the design equations for z and d. An interpolation routine was employed to transform from equal increments in ρ\ργ to equal increments in z. (The equal increments in z are more convenient for machining.) The n = 1 taper has a theoretical mode conversion that is less than approximately —33 dB for frequencies less than 12.83 GHz. The theoretical mode conversion for the n = 3 taper is less than approximately — 45 dB for frequencies less than 12.93 GHz. Measured values of mode conversion obtained with the n = 3 design [31] were less than —35 dB from 7 to 10 GHz. In this case, —35 dB represented the lower limit for accurate measurements; actual values of mode conversion were probably lower than this. The results obtained with the variable tapers should be compared with the theoretical ( < — 35 dB) and measured ( < — 30 dB) results obtained in the same frequency band with a 40-in. straight taper having the same input and output cross-sectional dimensions. * \A2\2 is the power in the spurious mode at the output of the taper for unit power incident in the desired mode.

196

JOHN P. QUINE TABLE IV DIMENSIONS FOR VARIABLE TAPER (n =

a, = 1.122 a2 = 2.800 z 0. 0.250 00 0.500 00 0.750 00 1.000 00 1.25000 1.500 00 1.750 00 2.00000 2.250 00 2.50000 2.75000 3.000 00 3.250 00 3.50000 3.750 00 4.000 00 4.250 00 4.500 00 4.750 00 5.00000 5.250 00 5.500 00 5.75000 6.000 00 6.25000 6.500 00 6.750 00 7.000 00 7.25000 7.500 00 7.75000 8.000 00 8.25000 8.500 00 8.75000 9.000 00 9.250 00 9.500 00 9.750 00 a

bl = 1.002 a = 0.9143 b2 = 2.500 Pl = 11.82 6/2 a/2

0.50098 0.503 86 0.512 17 0.525 06 0.54145 0.56029 0.580 70 0.602 03 0.623 77 0.645 60 0.66728 0.68865 0.709 61 0.730 10 0.75007 0.769 50 0.788 39 0.806 73 0.824 52 0.84178 0.858 51 0.874 73 0.890 45 0.905 68 0.920 44 0.934 74 0.948 60 0.962 02 0.975 02 0.987 62 0.999 82 1.01163 1.023 07 1.034 14 1.044 86 1.055 23 1.065 26 1.074 97 1.084 35 1.093 42

All dimensions are in inches.

0.56110 0.564 32 0.573 63 0.588 07 0.606 42 0.627 52 0.650 39 0.674 27 0.698 62 0.723 07 0.747 35 0.77129 0.794 77 0.817 71 0.840 08 0.861 84 0.883 00 0.903 54 0.923 47 0.942 79 0.96153 0.979 70 0.997 30 1.014 36 1.030 90 1.046 91 1.062 43 1.077 46 1.092 03 1.106 13 1.119 79 1.13302 1.145 83 1.158 24 1.170 24 1.18186 1.193 09 1.203 96 1.214 47 1.224 63

1) a

λ 0 = 0.915 ^=6.540 fQ = 12.93 Gc /i:2 = 0.766 z b/2 a/2

10.00000 10.250 00 10.500 00 10.750 00 11.000 00 11.25000 11.500 00 11.750 00 12.00000 12.250 00 12.50000 12.75000 13.000 00 13.250 00 13.50000 13.750 00 14.000 00 14.'250 00 14.500 00 14.750 00 15.00000 15.250 00 15.500 00 15.75000 16.000 00 16.25000 16.500 00 16.750 00 17.000 00 17.25000 17.500 00 17.75000 18.000 00 18.25000 18.500 00 18.75000 19.000 00 19.162 07 — —

1.102 18 1.110 64 1.118 81 1.126 69 1.134 28 1.14160 1.148 65 1.155 43 1.16194 1.168 20 1.17420 1.17996 1.185 46 1.190 72 1.195 74 1.200 53 1.205 08 1.209 39 1.213 48 1.217 34 1.22098 1.224 40 1.227 59 1.230 57 1.233 32 1.235 87 1.238 20 1.240 31 1.242 21 1.243 91 1.245 39 1.24667 1.247 73 1.248 59 1.249 24 1.24968 1.249 92 1.249 97 — —

1.23444 1.243 92 1.253 07 1.26189 1.270 40 1.278 59 1.286 49 1.294 08 1.30138 1.308 38 1.315 11 1.32155 1.327 72 1.333 61 1.339 23 1.344 59 1.349 69 1.354 52 1.359 10 1.363 43 1.367 50 1.37132 1.374 90 1.378 23 1.38132 1.384 17 1.386 78 1.389 15 1.39128 1.393 18 1.394 84 1.39627 1.397 46 1.39842 1.399 15 1.39965 1.399 91 1.399 96 — —

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TABLE V DIMENSIONS FOR VARIABLE TAPER {n = 3)fl

^ = 1,122 a2= 2.800 *

0. 0.250 00 0.500 00 0.750 00 1.000 00 1.250 00 1.500 00 1.750 00 2.000 00 2.250 00 2.500 00 2.750 00 3.000 00 3.250 00 3.500 00 3.750 00 4.000 00 4.250 00 4.500 00 4.750 00 5.000 00 5.250 00 5.500 00 5.750 00 6.000 00 6.250 00 6.500 00 6.750 00 7.000 00 7.250 00 7.500 00 7.750 00 8.000 00 8.250 00 8.500 00 8.750 00 9.000 00 9.250 00 9.500 00 9.750 00 10.000 00 a

bl = 1.002 6 2 = 2.500 6/2 0.500 98 0.501 01 0.501 40 0.503 01 0.507 05 0.514 72 0.526 77 0.543 26 0.563 61 0.586 87 0.612 05 0.638 28 0.664 91 0.691 46 0.717 59 0.743 10 0.767 86 0.791 77 0.814 80 0.836 94 0.858 18 0.878 53 0.898 03 0.916 68 0.934 51 0.951 57 0.967 86 0.983 43 0.998 30 1.012 50 1.026 05 1.038 98 1.051 32 1.063 08 1.074 30 1.084 99 1.095 17 1.104 86 1.11409 1.122 86 1.131 20

All dimensions are in inches.

a = 0.9143 p x = 11.82 fl/2 0.561 10 0.561 13 0.561 56 0.563 37 0.567 90 0.57649 0.589 99 0.608 46 0.631 24 0.657 29 0.685 49 0.714 88 0.744 70 0.774 43 0.803 70 0.832 27 0.860 00 0.886 78 0.912 58 0.937 37 0.961 16 0.983 96 1.005 79 1.026 68 1.046 65 1.065 75 1.084 00 1.101 44 1.11809 1.134 00 1.149 18 1.163 66 1.177 48 1.190 65 1.203 22 1.215 19 1.226 59 1.237 45 1.247 78 1.257 60 1.266 94

λ 0 = 0.915 ΑΊ = 6.540 f0 = 12.93 Gc K2 = 0.766 z A/2 a/2 12.500 00 12.750 00 13.000 00 13.250 00 13.500 00 13.750 00 14.000 00 14.250 00 14.500 00 14.750 00 15.000 00 15.250 00 15.500 00 15.750 00 16.000 00 16.250 00 16.500 00 16.750 00 17.000 00 17.250 00 17.500 00 17.750 00 18.000 00 18.250 00 18.500 00 18.750 00 19.000 00 19.250 00 19.500 00 19.750 00 20.000 00 20.250 00 20.500 00 20.750 00 21.000 00 21.250 00 21.500 00 21.750 00 22.000 00 22.250 00 22.500 00

1.193 99 1.198 50 1.202 75 1.206 73 1.210 46 1.213 96 1.217 22 1.220 26 1.223 08 1.225 72 1.228 15 1.230 40 1.232 48 1.234 39 1.236 15 1.237 75 1.239 22 1.240 55 1.241 76 1.242 85 1.243 83 1.244 70 1.245 48 1.246 18 1.246 79 1.247 32 1.247 79 1.248 19 1.248 53 1.248 82 1.249 07 1.249 27 1.249 44 1.249 57 1.249 68 1.249 76 1.249 83 1.249 88 1.249 91 1.249 93 1.249 95

1.337 27 1.342 33 1.347 08 1.351 54 1.355 72 1.359 63 1.363 28 1.366 69 1.369 85 1.372 80 1.375 53 1.378 05 1.380 38 1.382 52 1.384 48 1.386 28 1.387 92 1.389 42 1.390 77 1.391 99 1.393 08 1.394 07 1.394 94 1.395 72 1.396 40 1.397 00 1.397 52 1.397 97 1.398 35 1.398 68 1.398 96 1.399 18 1.399 37 1.399 52 1.399 64 1.399 74 1.399 81 1.399 86 1.399 90 1.399 92 1.399 94 (continued)

JOHN P.. QUINE

198

TABLE V (Continued) »ι = 1 ,122 a2 = 2,800 z 10.250 10.500 10.750 11.000 11.250 11.500 11.750 12.000 12.250 a

00 00 00 00 00 00 00 00 00

bi = 1.002 b2 = 2.500

a = 0.9143 p,= 11.82

bll

all

1.139 1.146 1.153 1.160 1.166 1.172 1.178 1.184 1.189

11 63 76 52 92 97 69 09 19

1.275 1.284 1.292 1.299 1.306 1.313 1.320 1.326 1.331

81 22 21 78 95 73 13 19 89

λ 0 = 0.915 ATi = 1.540 / o = 12.93 Gc K2 = 0.766 b/1 z all 22.750 23.000 23.250 23.500 23.750 23.964

— — —

00 00 00 00 00 99

1.249 1.249 1.249 1.249 1.249 1.249

— — —

96 96 96 97 97 97

1.399 1.399 1.399 1.399 1.399 1.399

— — —

95 96 96 96 96 96

All dimensions are in inches.

Reflection from these tapers is small, and the VSWR is typically less than 1.04. provided the standard waveguide is operating well above cutoff occurring at ax/λ = 0.5.High reflections can occur at the throat (YSWR > 1.1) for αι/λ less than approximately 0.65. Measurements by Younger [46] show that straight tapers have lower reflections than variable tapers of comparable length B. DESIGN OF BENDS

1. Gradual Bends A gradually bent waveguide can be represented as a set of n continuously coupled transmission lines, n being the number of propagating modes. The relations among the forward-traveling normalized coupled-mode amplitudes, An, are given by the coupled transmission line equations: dA, —± + βχ A, + c12 A2 + · · · + clnAn = 0, dz dA -^ + cl2Al+ß2A2 dz

+ --< + c2nAn = 0,

dA -^ + clnAl+c2nA2 dz

+ '-' + ßnAn

(28)

=0.

Reflected waves are neglected, since these " phase out" in a bend of reasonable length. In (28), ßn is the propagation constant of the nth coupled mode, and

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cmn is the amplitude coupling coefficient per unit length between modes m and n. The distance z is measured along the centerline of the bend. Relatively simple solutions of (28) have been obtained by Miller [3] for constant curvature bends in circular waveguides carrying the ΤΕ£Ί mode by considering only the TE£i mode and the degenerate TMfi spurious mode (ßi = ßi)- These two-mode solutions show that complete power transfer occurs cyclically between the two modes along the length of the bend, and that zero energy in the TMp! mode occurs at the output of the bend for certain discrete bend angles given by sn [λ\, 1.16 \aj

,χ

sn c12R

(29)

where s is any integer, a is the waveguide radius, R is the bend centerline radius, λ is the free space wavelength, and cl2 is the amplitude coupling coefficient per unit length between the TEjpi and TMft modes. The insertion loss for these discrete bends which results from energy conversion from the TE^i to the TMp! mode is L(dB) = 42.8(5Δ/1//1)2,

(30)

where Αλ is the difference between the operating wavelength and the design center wavelength, λ. Note that 9S and L given by (29) and (30), which were obtained from the two-mode solution of (28), are independent of R. However, power is also coupled cyclically to the nondegenerate spurious modes, principally to the TEpj and TEp 2 modes. Thus, the total mode conversion loss is given by (30) only for large values of R/a; in this case, the power in the nondegenerate modes can be neglected. The minimum value of R/a can be estimated by calculating the maximum power, Pn, that can be coupled to the nth spurious mode for unit power incident in the TEfi mode. Miller [17] has shown that Pn is equal to the inverse of 1 + Fn2, where

and the subscripts 1 and n refer to the desired T E ^ mode and any one of the spurious TE2, or TM£ n modes, respectively. For example, far above cutoff, cl2 = \.93(a/RX) and F2 = 0.285(/?/a) for the TEp2 mode. Thus, R/a must have a value of approximately 35 for P2 = 0.01. Bends can also be designed for circular waveguides carrying the TEjpi mode by employing techniques for removing the degeneracy between the TE^i and T M P modes. This can be accomplished through the use of elliptical or corrugated circular waveguides [3], or dielectric coated circular waveguides

200

JOHN P. QUINE

[32]. It has also been shown that the T E $ mode can be made the dominant mode in circular waveguide by the proper choice of surface reactance [33,34]. With the TE9i, TMft degeneracy removed, low loss bends of any angle, 0, can be obtained by employing a sufficiently large value of R/a, to result in large values of Fn for all modes. Dimensions for bends having low mode conversion loss have also been determined for rectangular waveguides carrying the TE° 0 mode [20,35]. In this case, more compact bends are possible, since the ΤΕ? 0 mode is degenerate with no other mode. For example, a compact constant curvature //-plane bend is possible having R/a = 1.48, where R is the average bend radius and a is the waveguide width. Measured values of the powers in the TE° 0 , TE° 0 , and TEJ 0 spurious modes at the output of the bend were below — 20 dB relative to the power incident in the TE° 0 mode for all frequencies corresponding to α/λ in the range between 1.3 to 2.2, and were below — 28 dB over approximately 5 % bandwidths centered at frequencies corresponding to α/λ = 1.4 and 2.0. Spurious mode powers less than — 30 dB can be obtained over broader bandwidths with bends having variable radii of curvature. 2. Quasi-Optical Bends Figure 5 shows a miter bend based on quasi-optical principles. Reasonably low mode conversion loss can be obtained with such bends, provided the waveguide cross-sectional dimensions are large compared to the wavelength. Thus, the miter bend is useful for greatly oversize waveguide systems. The chief advantage of the miter bend is its greater compactness and bandwidth compared to the gradual bends discussed previously. Approximate formulas have been derived for the total mode conversion loss in quasi-optical bends in circular waveguides carrying the TE^i mode and in rectangular waveguides carrying the ΤΕ? 0 mode. The total mode conversion loss for the TE° 0 mode in a quasi-optical £-plane bend in rectangular waveguide of height, b (T = b in Fig. 5), is given by [36] / λ \1/2 L(dB)=1.96—— .

(32)

For a quasi-optical //-plane bend in rectangular waveguide of width, a(T = a in Fig. 5), the total mode conversion loss for the TE° 0 mode is given by [36] / λ \3/2 L(dB) = 2.05 - ^ — . (33) \a sin Θ] The total mode conversion loss for the T E ^ mode in a quasi-optical bend in a circular waveguide of radius, a (T = 2a in Fig. 5), is given by [23,36] / λ \3/2 L(dB) = 2 . 4 — — . (34) \a sin 0/

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201

Note that L(dB) is higher for an £-plane than for an //-plane bend, and that the mode conversion loss is smallest for a bend angle 0 = 90°. The results of experiments [37] show that Eqs. (32)-(34) provide accurate results for values of μ = [λ/lTsm Θ] which are less than approximately 0.194. Thus, the projected aperture, Τύηθ, must be large compared to λ in order for the quasi-optical results to apply. For larger values of μ (longer wavelengths) measured values of mode conversion loss are less than the calculated values. 0/2

~T~T T

_J

T

FIG. 5. Quasi-optical bend. C. DESIGN OF M U L T I H O L E DIRECTIONAL COUPLERS

The design of multihole directional couplers for standard-size waveguides has been discussed by many authors [38,39-42]; only couplers for oversize waveguides for high-power applications will be discussed here. Two major problems that arise are the problem of obtaining sufficient coupling per unit length for the desired mode, and the problem of obtaining low spurious mode amplitudes at the output ports. Desired mode coupling is inherently low in oversize waveguide couplers, because of the reduced electric and magnetic fields at the waveguide walls for a given power incident in the desired mode. Spurious mode suppression in multihole couplers can be obtained by employing a coupling length, L, which is large relative to the beat wavelength, 2 b e a t , between the desired mode and the spurious mode [17]; in this case, the suppression is greater, the greater the ratio, L/Abeat. However, far above cutoff the values of 2 beat for the lower-order spurious modes become very large, and very large values of L are therefore required for spurious mode suppression. For simplicity, the discussion will be restricted to rectangular waveguide couplers for the TE° 0 . Many of the conclusions reached, however, will apply equally well for circular waveguide couplers for either the TEft or TEfi mode.

202

JOHN P. QUINE

1. Coupling through Small Holes The polarizability concept [43,44] can be employed to calculate the coupling through single holes whose critical dimensions are small compared with the half-wavelength, A/2. The polarizabilities have been determined for holes of various shapes [44,45]. As an example, the case of sidewall coupling between two rectangular waveguides coupled through a uniform grating of metallic strips as shown in Fig. 6 will be considered. The waveguides have equal

FIG. 6. Multihole sidewall coupler.

heights, b, and widths, ax and a2. The grating slots have width, vr, length, b, and spacing, d; the grating strips have width, s. The amplitude coupling coefficient per slot, kpJ^u between the TE° 0 mode in waveguide number 1 and the TE° 0 mode in waveguide number 2 is independent of b and is given by* [2,31,44] _ n2mpTw2 (Âm0Xp0\1/2 km01

"

32

\a~WJ

(55)

In (35), Xm0 and λρ0 are the wavelengths of the TE° 0 and TE° 0 modes; it is assumed that these modes are not too close to cutoff. T is a correction factor for the finite wall thickness, /, and is given by [44]

τ

=Η-νί(τ) -1] )·

(36)

Equation (35) should be employed only for weak coupling (s/d not too close to zero) and for d/λ considerably less than 0.5. For strong coupling (s/d approaching zero) or for d/λ approaching 0.5, more accurate results can be obtained by a transverse resonance method [41] employing the plane wave reactance of the grating [2]. Equation (35) shows that, when operation is far above cutoff (λη0 and λρ0 approach λ), the coupling is inversely proportional to (ala2)3/2. This * The power coupled to mode/? is |££oi| 2 for unit power incident in mode m.

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203

rapid decrease in coupling with increasing waveguide width results from the decrease in the longitudinal magnetic field along the coupling wall. Equation (35) also shows that the ratio of the amplitude of the spurious TE° 0 mode to the amplitude of the TE° 0 mode produced in waveguide number 2 by an incident TE° 0 mode in waveguide number 1 is LP022 MOI K

101

pO

1/2

L^IO

(37)

Thus, the coupling from each slot to the spurious TE° 0 mode is more than p times the coupling to the desired TE° 0 mode. This increases the problem of suppressing the spurious modes. Letting m = p in (35), the following ratio can be derived: ^ρθ

K

102 ~" P ioi L^ioJ

(38)

Equation (38) shows that the coupling between TE° 0 modes is more than p2 times the coupling between TE° 0 modes. This effect can introduce errors in the measurement of the mode purity at the coupled port of loosely coupled directional couplers, if a source having finite mode purity is employed [22,31]. 2. Methods for Obtaining Flat Coupling Γη the case of top-wall coupling between two oversize rectangular waveguides, flat coupling, i.e., coupling characteristics independent of frequency, can be obtained by employing the branch-guide principle [39,42]. Since the coupling slots in this case extend entirely across the waveguides, the only spurious modes generated by an incident TE° 0 mode are the T E ° and T M ° modes; these combine to form the LSE ln modes [9,11] having transverse electric field parallel to that of the incident TEjo mode. Three-decibel branchguide couplers employing six branches have been built recently [46] in "flat" oversize waveguides having a = 0.203 m (8.0 in.) and b = 0.034 m (1.34 in.). The unbalance between output ports was only +0.2 dB over a 19 % bandwidth centered at 3.35 GHz. With full-oversize waveguides, i.e., b & a, inconveniently long coupling lengths may be required to suppress the LSE n mode; in this case also, the branch-guide heights may become inconveniently small because of the large number employed. Flat coupling can also be obtained between two oversize rectangular waveguides coupled along a common side wall. For this case, Fig. 7 shows a typical plot of the total coupling from an array of Wholes as a function of the frequency parameter, α/λ, where a is the waveguide width and λ is the free space wavelength. In the case of the TE° 0 mode, coupling occurs only through the longitudinal magnetic field, Hz. For small values of α/λ, (35) applies, and

204

JOHN P. QUINE ML· 102 37Γ MU 102 - IL ioi " 2 /^ toi ~ 2

rgK

3.0 3.2

3.5

FIG. 7. Typical frequency dependence of the total coupling for multihole sidewall couplers.

this equation shows that the coupling per hole, k\%\, is proportional to A10. Thus, k\°Q\ = Κλί0, where K is a constant. The total coupling with N identical holes is approximately sm(Nk\%\) [17]. This results in the rapid oscillations of the total coupling for small values of α/λ as shown in Fig. 7. For larger values of α/λ, the value of Nk\%\ decreases, and the oscillations cease. When Nk\°0\ is equal to π/4, a total coupling of - 3 dB is obtained. If the coupling per hole, k\%\, were proportional to λχο for indefinitely large values of α/λ, the total coupling, equal to sin(N/C}oi)> would approach ΝΚλί0. However, when the critical hole dimensions become comparable to A/2, resonance effects are experienced; this causes the coupling to depart from ΝΚλιο and eventually to increase with increasing frequency as shown in Fig. 7. The result can be a rather broad frequency range over which the coupling is relatively independent of frequency. A second effect associated with the " grating effect" [47-49], occurs at the critical frequency,/,., corresponding to a hole spacing, d, equal to approximately λί0/2.* Τη this case, λί0 is the wavelength * Actually, the grating effect occurs for an angle of incidence, θ0 [41], given by sin #o = (λ/ί/) — 1, and corresponds to the first appearance of the reflected wave of order r = — 1 at an angle #_ i = — 90°. This occurs at a slightly lower frequency than the frequency fc corresponding to λ ί 0 = 2d. Atfc, the reflected waves for r = 0 and — 1 have equal amplitudes but opposite angles, and result in a perfect standing wave distribution for the even characteristic TEfo mode along the waveguide axis. High reflection occurs in this case.

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of the even characteristic mode of the pair of coupled waveguides corresponding to the TEfo mode of the unperturbed waveguides. At/ C , a sharp discontinuity occurs in the slope of the coupling characteristics as shown in Fig. 7. In the oversize waveguides, A10, as defined previously, is only slightly greater than λ. Flat coupling characteristics (±0.20 dB over a 15% frequency band) have been obtained experimentally with multihole sidewall couplers for oversize rectangular waveguides having total coupling values ranging between —3 and —22.5 dB [22]. These couplers also exhibited low reflection, high directivity, and low mode conversion loss at frequencies less than approximately 0.95 fc. 3. Calculation of the Spurious Mode Amplitudes The problem of calculating the spurious mode amplitudes at the output ports of a multihole coupler can be simplified by considering the characteristic modes of the pair of coupled waveguides in the limit of very weak coupling. The characteristic modes can be classed as "even" or "odd," depending on the symmetry of the transverse fields with respect to the plane of the coupling wall [31,41,42,50]. If the waveguides are identical, the excitation of only one of the waveguides corresponds to an equal excitation of the even and odd characteristic modes. If the coupling wall is infinitely thin, the odd modes are not affected by the presence of the coupling holes of finite size. In this case, the odd modes form an uncoupled set, as in the limit of very weak coupling, and the odd component of the TEfo mode in passing through the coupler produces no spurious modes, even or odd. The even modes, on the other hand, are coupled to each other by the coupling holes but not to the odd modes. The even component of the TEPo mode in passing through the coupler experiences coupling to all of the even spurious modes. Since only the even spurious modes are excited in the case of infinitely thin walls, it can be seen that the spurious mode amplitudes at the output ports of the coupler are identical in magnitude and phase. Relatively simple expressions can be derived for the amplitudes of the spurious modes at the output ports of the coupler by employing the loose coupling approximation. In this approximation, it is assumed that the total power coupled to the spurious modes is small. For loose coupling, one can also neglect the perturbations of the even mode propagation constants caused by the holes [17,31]. Under these conditions, the even spurious mode amplitude, Afmn{L), at both forward output ports is given by AUL) = exp(-jßmnL) Σ *7δΪω πρί-Kßio - /U*l· 2=0

(39)

206

JOHN P. QUINE

and the even spurious mode amplitude, Abmn(0), at both backward ports is given by Abmn(0) = Σ *7Sfc) exp[-;(/? 1 0 + / U z ] ·

(40)

2= 0

Equations (39) and (40) represent the summations of the spurious even mode contributions from each of the coupling holes in the array of TV holes having constant spacing, d. The axial coordinate, z, takes on only discrete values equal to multiples of d. L = (N- \)d is the length of the array, and ßmn is the modal propagation constant = 2π/ληιη. In the case of sidewall couplers, (37) can be used to relate the spurious mode coupling coefficient, &n>i(z), to the desired mode coupling coefficient k\%\{z). Relations similar to (37) can be derived for top wall couplers. Thus, the spurious mode amplitudes can be calculated, if the desired mode coupling coefficients are specified. Miller [17] has defined the normalized variable 0 ^ = LM 1 o±LM IBO = LM beat ,

(41)

where the plus and minus signs refer to backward- and forward-traveling waves, respectively. Use of (37) and (41) in (39) results in the following equation in the case of the sidewall coupler : AUL)

= m(jff2

expi-j/UD Σ/Γοΐω

ε χ

ρ(-^)>

( 42 )

and similarly for Abm0(0). If&Joi(z) = C a constant, the magnitude of Afm0(L) can be expressed in the following form : \Afm0(L)\ = sin(NC)

qmo\1/2( NC \ sinlMl(N - 1)] U10/ Isin(TVC)/ Nsin[ß/(N-l)"]'

{

)

Since sin(iVC) is the total coupling for the desired TEfo mode [17], it is seen that the left-hand side of (43) is equal to the inverse of the mode purity at the coupled output port. Inspection of the right-hand side shows that the mode purity is a relatively slowly varying function of NC even for values of NC as high as π/4 (3-dB coupler). Equation (43) shows that the spurious mode amplitude is zero when θ/π is an integer, or L = (N — \)d is a whole number of beat wavelengths. Spurious mode maxima or "side lobes" occur when θ/π is approximately (m + ^), where m is an integer. The forward-traveling spurious modes usually correspond to values of θ/π on the order of unity [minus sign in (41)]. In this case, the envelope of the mode purity is approximately proportional to (ö/π), since sm[6/(N — 1)] in the denominator of (43) can be replaced with Θ/(Ν - 1)

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when TV is large. Thus, the mode purity increases slowly with increasing θ/π when a uniform coupling distribution is employed. In this case, very long coupling lengths must be employed, if high mode purity is required over broad frequency bandwidths. The use of tapered coupling distributions [22,31] can result in high mode purity with reduced coupling lengths. Mode purity > 26 dB with θ/π > 2.0 has been obtained with sin2 πζ/L coupling distributions. Equation (43) shows that the mode purity is a periodic function of (θ/π) with period equal to (N — 1). This is true for any coupling distribution if the spacing, d, is a constant. Very high values are obtained for the backward wave amplitude when θ/π is equal to TV — 1. This results from an in-phase addition of the backward wave contributions from all the holes, and can be avoided for all backward waves if the spacing, d, is made less than λ10/2. The spacing can be quite close to A10/2, however, if the coupling length, L, is made sufficiently large (see foregoing discussion on methods for obtaining flat coupling). D.

DESIGN OF QUASI-OPTICAL COUPLERS

Figure 8 shows a quasi-optical — 3-dB hybrid coupler that has approximately the same mode conversion loss [Eqs. (32)-(34)] as the quasi-optical

ÎO

\

1.0 ►

!

,,

±J V2

\ |

\

\

J_ V2

Coupling Wall

FIG. 8. Quasi-optical coupler.

bend (Θ = 90°) shown in Fig. 5. The quasi-optical coupler, therefore, shows considerable promise for application in the greatly oversize waveguide systems required for super-high-power transmission.

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JOHN P. QUINE

Any degree of coupling can be obtained (and apparently with nearly the same mode purity) by varying the transmission coefficient of the coupling wall. One type of coupling wall that has been employed for low-power millimeter waves [51,52] consists of one or more dielectric sheets. For high average power, the wall can be cooled by a liquid dielectric, as has been done successfully in the case of windows for high-power microwave tubes. For small coupling values, a perforated metal sheet surrounded by dielectric coolant can be employed. The spacings of the perforations should be small enough to avoid grating lobes [47,48]. Since the angle of incidence on the coupling wall is approximately 45°, the spacing should be less than approximately 0.586/1. E. ADDITIONAL COMPONENTS

Several additional components for high-power oversize waveguide systems have been developed or shown to be feasible. These include mode-selective absorbers, transducers from rectangular to circular waveguides, rotary joints, switches, and duplexers. Mode-selective absorbers that provide high absorption for the spurious modes and negligible loss for the desired mode are required to damp out the effects of trapped mode resonances [see Eqs. (5)-(7)]. In the case of circular waveguides, a length of helix waveguide [53] can be inserted to absorb all modes except those belonging to the ΊΕ$η mode family; the TE$, modes are not absorbed, because these have no longitudinal currents. For high-power applications, the spaced-ring waveguide [3] may have advantages over the helix waveguide. Mode-selective absorbers have also been developed for rectangular waveguides [22]. The configuration developed for selectively absorbing the TE° 0 mode is a multihole sidewall directional coupler consisting of a main waveguide of width, a, coupled symmetrically on each sidewall to waveguides of width approximately aß. The TE^ 0 mode in the main waveguide is degenerate with the TE° 0 mode in the side waveguides, and is therefore strongly coupled. An X-band model produced approximately 6 dB insertion loss for the spurious TE° 0 mode, and approximately 0.03 dB insertion loss for the desired TE° 0 mode. Even lower TE° 0 mode losses should be possible by a better fabrication method, e.g., by electroforming. The design of a mode-selective absorber for the TE°„ and TM°„ modes presents a special problem, because these modes are degenerate and can combine to form LSM and LSE modes [9,11]. The LSM modes, like the desired TE° 0 mode, have zero longitudinal currents on the sidewalls, and therefore cannot be absorbed selectively by narrow transverse slots on these walls. Only the LSE modes having electric field normal to the side walls can be absorbed by such slots. Furthermore, the slots do not couple the LSE and LSM modes.

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For this reason some means must be employed to provide coupling from the LSM to the LSE modes. This can be accomplished by deforming the waveguide from the perfect rectangular shape in order to cause the propagation constants of the TE^n and TM°„ modes to become unequal; that is, the degeneracy between these modes is removed. Meinke has proposed an optimum waveguide for this purpose having slightly curved top and bottom walls, as shown in Fig. 5 of Meinke et al. [54]. This optimally-shaped waveguide can be closely approximated, however, by an hexagonal waveguide. It has also been shown [22] that, if the trapping of the spurious mode occurs between gradually tapered transitions from standard to oversize waveguides, the tapers provide the optimum coupling between the LSM and LSE modes. In this case, absorption of both of these modes can be obtained by transverse slots on the side walls of the rectangular waveguides. Several configurations are possible for transitions from the TE° 0 mode in standard size rectangular waveguide to the TE£\ mode in circular waveguide [55,56]. A transition from the TE° 0 mode in oversize rectangular waveguide to the T E ^ mode in circular waveguide has also been proposed [22]. A rotary joint for a rectangular waveguide system can be obtained by employing a pair of such transitions. In this case, the motional joint is placed in the circular waveguide section carrying the TE^i mode, since this mode has no longitudinal currents. A spark-gap switch has been developed for oversize rectangular waveguides [57]. This was patterned after a switch for standard size waveguides [58] containing a single dc-triggered spark gap. The oversize waveguide switch employs a symmetrical array of eight dc-triggered spark gaps in order to minimize the generation of spurious modes. Duplexers have also been shown to be feasible for oversize rectangular waveguides, and are currently under development [46]. IV. Conclusions

The factors affecting waveguide attenuation due to finite wall conductivity and mode conversion, and the factors affecting waveguide peak and average power-handling capabilities have been reviewed. The attenuation can be decreased by orders of magnitude, and the peak and average power-handling capabilities can be increased by orders of magnitude through the use of oversize waveguides. These advantages can be realized, however, only if the spurious propagating modes are controlled by the careful design of components to minimize mode conversion, and by employing mode-selective absorbers to damp out spurious mode resonances. Two classes of components have been described. The first class has moderately oversize cross-sectional dimensions (1.5 to 2.5A), and can carry more than an order of magnitude higher power than standard-size components.

210

JOHN P. QUINE

Components in the second class are designed by quasi-optical principles, and therefore operate properly only for greatly oversize cross-sectional dimensions (10A or greater). These components can carry several orders of magnitude higher power than standard-size components. Components in both classes can be designed to have acceptably low mode conversion losses (0.011 dB or lower, corresponding to spurious mode amplitudes less than — 26dB). ACKNOWLEDGMENTS

The writer is indebted to Mr. V. C. Vannicola of the Rome Air Development Center for encouraging and providing support for investigations of oversize waveguides and components, to his colleague Mr. Cousby Younger for many discussions, and to Dr. Georg Goubau for his review of the manuscript. SYMBOLS

C

D d d,dlt ^max

/

fc F(oc x) Fn

H KuK2

Km

L·

•^nm

L LSE,M

Normalized attenuation, Eq. (1) Absorption in decibels experienced by spurious mode Normalized mode amplitudes Radius of circular waveguide Width and height of rectangular waveguide Normalized amplitude coupling coefficient Coupling factor for variable taper Amplitude coupling coefficient per unit length Diameter of circular waveguide Coupling hole spacing Cross-sectional dimensions in variable taper Peak breakdown electric field Frequency Critical frequency for 2d= λ 1 0 Factor for variable tapers, Eqs. (11)-(14) Coupling parameter, Eq. (31) Heat-transfer coefficient, Eq. (2) Factors for variable taper, Eq. (Π) Coupling per slot, Eq. (35) Roots ofVi(A;») = 0 Roots of Jm(knm) = 0 Length of straight taper, length of multihole coupler Longitudinal section electric, magnetic modes

m, n N p

1

max

P R Rs S s T

τ T

w w a a

βΜ,β. ßo V

λ

κA

mB Abeat

^

ΘΛ θ/π

P>pi Or

Mode indices Number of coupling holes Peak breakdown power Mode index Bend centerline radius Waveguide surface resistance Waveguide surface area per foot, Eq. (2) An integer, Eq. (29) Waveguide cross-sectional dimension, Figs. 2 and 5 Thickness correction for coupling, Eq. (36) Waveguide temperature, Eq. (2) Watts dissipated per foot, Eq. (2) Width of coupling slots Attenuation per unit length Parameter, Eqs. (12)—(14) Modal propagation constants Propagation constant of free space Impedance of free space Free space wavelength Waveguide wavelength Wavelength of mode mn Beat wavelength Parameter for quasi-optical bend, Eqs. (32) and (34) Bend angle Normalized variable, Eq. (41) Phase factors for variable taper, Eq. (11) Conductivity relative to copper

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References

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23. E. A. J. Marcatili, "Miter elbow for circular electric mode," Proc. Symp. QuasiOptics, pp. 535-543. Polytechnic Press, New York, 1964. 24. L. B. Felsen, "Quasi-optical methods in microwave propagation and diffraction," AppL Opt., vol. 4, pp. 1217-1228, October 1965. 25. L. Solymar, "Design of a conical taper in circular waveguide system supporting H 0 i mode," Proc. IRE, vol. 46, pp. 618-619, March 1958. 26. L. Solymar, "Design of a two-section conical taper in circular waveguide system supporting the H 0 i mode," Proc. 1EE (London), Pt. B, vol. 106, Suppl. (convention on long-distance transmission by waveguide), pp. 119-120, January 1959. 27. L. Solymar, "Monotonie multi-section tapers for over-moded circular waveguides," Proc. IEE (London) Pt. B, vol. 106, pp. 121-128, January 1959. 28. L. Solymar, "Step transducer between over-moded circular waveguides," Proc. IEE (London) Pt. B, vol. 106, pp. 129-131, January 1959. 29. L. Solymar, " Mode conversion in pyramidal-tapered waveguides," Electron. Radio Engr. (London), vol. 36, pp. 461-463, December 1959 (right-hand side of Eq. (5) should be multiplied by L). 30. C. C. H. Tang, "Optimization of waveguide tapers capable of multimode propagation," IRE Trans. Microwave Theory Tech., vol. MTT-9, pp. 442-452, September 1961. 31. J. P. Quine, C. Younger, and J. W. Maurer, " U l t r a high power transmission line techniques," RADC-TR-65-164, Final Rept. Contract AF30(602)-2990, September 1965. 32. H. G. Unger, " N o r m a l mode bends for circular electric waves," Bell System Tech. J., vol. 36, pp. 1292-1307, September 1957. 33. H. E. M. Barlow, " A method of changing the dominant mode in a hollow metal waveguide and its application to bends," Proc. IEE (London), Pt. B, vol. 106, Suppl. (convention on long distance transmission by waveguide), pp. 100-105, January 1959. 34. J. B. Davies, "An investigation of some waveguide structures for the propagation of circular TE modes," Proc. IEE (London), Pt. C, vol. 109, pp. 162-171, March 1962. 35. M. G. Andreasen, "Synthesis of a bent waveguide with continuously variable curvature," Arch. Elec. Übertr., vol. 12, pp. 463-471, October 1958. 36. B. Z. Katsenelenbaum "Diffraction on plane mirror in broad-waveguide junction," Radio Eng. Electron. Phys. (USSR) (English Transi.), vol. 8, pp. 1098-1105, July 1963. 37. R. B. Vaganov, " Measurement of losses in certain quasi-optical waveguide elements, Radio Eng. Electron. Phys. (USSR) (English Transi.), vol. 8, pp. 1228-1238, July 1963. 38. S. E. Miller and W. W. Mumford, " Multi-element directional couplers," Proc. IRE, vol. 40, pp. 1071-1078, September 1952. 39. J. R. Reed, " T h e multiple branch waveguide coupler," IRE Trans. Microwave Theory Tech., vol. MTT-6, pp. 398-403, October 1958. 40. W. E. Caswell and R. F. Schwartz, " T h e directional coupler—1966," IEEE Trans. Microwave Theory Tech., vol. MTT-15, pp. 120-123, February 1967. 41. K. Tomiyasu and S. B. Cohn, " T h e transvar directional coupler," Proc. IRE, vol. 41, pp. 922-926, July 1953. 42. R. Levy, "Directional couplers," Advan. Microwaves (L. Young, ed). New York: Academic Press, vol. 1, pp. 115-209, 1966. 43. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev., vol. 66, pp. 163-182, October 1944.

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